PENELITIAN OPERASIONAL I (TIN 4109)

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1 PENELITIAN OPERASIONAL I (TIN 4109)

2 Lecture 2 LINEAR PROGRAMMING

3 Lecture 2 Outline: Introduction to Linear Programming Graphic Method References: Frederick Hillier and Gerald J. Lieberman. Introduction to Operations Research. 7th ed. The McGraw-Hill Companies, Inc, Hamdy A. Taha. Operations Research: An Introduction. 8th Edition. Prentice-Hall, Inc, Sankaranarayanan, Sriram. Lecture Note. University of Colorado Boulder, 2013.

4 Definition and Example Allocating limited resources among competing activities in a best possible (i.e., optimal) way. Selecting the level of certain activities that compete for scarce resources that are necessary. How much each resouce will be consumed by each activity. Uses a mathematical model to describe the problem of concern. Linear: all mathematical functions in the model must be linear functions. Programming: planning. Planning activities to obtain an optimal result, i.e., a result that reaches the specified goal best (according to mathematical model) among all feasible alternatives.

5 Definition and Example

6 Definition and Example Basic components: 1. Decision variables that we seek to determine. 2. Objective (goal) that we need to optimize (maximize or minimize). 3. Constraint that the solution must satisfy. Steps: 1. Proper definition of the decision variables 2. Constructing the objective function and constraints 3. Solving the problem

7 Definition and Example Example of LP problem: Perusahaan kaca WYNDOR memproduksi kaca (jendela dan pintu) dengan kualitas tinggi. Perusahaan ini mempunyai tiga departemen. Departemen 1 membuat rangka alumunium dan perkakas logam, Departemen 2 membuat rangka kayu, dan Departemen 3 membuat kaca dan merakit sebuah produk. Akibat penurunan pendapatan, pihak atasan memutuskan untuk mengubah lini produk perusahaan. Produk yang tidak mendatangkan keuntungan akan dihentikan dan perusahaan akan menentukan kapasitas produk untuk membuat dua produk baru yang dinilai mempunyai potensi pasar tinggi. Produk 1: pintu kaca dengan rangka aluminium berukuran 8 kaki Produk 2: rangka rangkap jendela dari kayu berukuran 4 x 6 kaki Produk 1 membutuhkan proses di Departemen 1 dan 3. Produk 2 diproses pada Departemen 2 dan 3. Perusahaan memiliki data sebagai berikut: Departemen Waktu Prod / Batch (Jam) Produk 1 Produk 2 Waktu Prod yg tersedia / minggu (Jam) Define the variables Keuntungan/batch $3000 $ Construct the objective function and constraints

8 Graphic Method Example of LP

9 Graphic Method Selesaikan permasalahan contoh soal perusahaan WYNDOR dengan menggunakan metode grafik.

10 Visualizing General Form of LP:

11 Visualizing Feasible Region:

12 Visualizing Optimization

13 Visualizing Solving Linear Problems: Outcome #1: Optimal solution(s) exists Outcome #2: Objective function is unbounded Outcome #3: Feasible region is empty

14 Visualizing Unbounded Problem (Example):

15 Visualizing Infeasible Problem: Issue: Constraints contradict each other.

16 Visualizing Solving Linear Problems: 1. Find which of three cases are applicable. Infeasible? Unbounded? Feasible + Bounded = Optimal? 2. If Optimal, find optimal solution. Note multiple optimal solutions possible

17 Properties Proportionality Contribution of each decision variable in both objective function and the constraints to be directly proportional to the value of the variable. Additivity Total contribution of all the variables in the objective function and in the contraints to be the direct sum of the individual contributions of each variable. Certainty All the objective and contraints coefficients of linear programming model are deterministic. Divisibility Desicion variables in a linear programming model are allowed to have any values, including noninteger values, that satisfy the functional and nonnegativity contraints.

18 Algorithms Solving systems of Linear Ineaqualities Early work by Fourier (Fourier-Motzkin Elimination Algorithm). Linear Arithmetic. World War II: Optimal allocation of resources. Advent of electronic / mechanical calculating machines. L.V. Kantorovich in USSR (1940) and G.B. Dantzig et al. In the USA (1947).

19 Algorithms Simplex. Ellipsoidal Methods. Interior Point Methods.

20 LP: Exercises Buat formulasi LP dari permasalahan berikut: NORI & LEETS CO., salah satu produsen baja utama di dunia, bertempat di kota Steeltown dan satusatunya perusahaan yang mempunyai banyak pekerja. Perusahaan memiliki dua sumber polusi utama: Blast Furnaces (BF) dan Open-Heart Furnaces (OF). Perusahaan ingin memperbaiki kualitas lapisan udara di kota Steeltown dengan mengurangi polutan yang mereka timbulkan. Adapun data yang dimiliki adalah sebagai berikut: Polutan Laju Pengurangan pada Emisi Tahunan yg diperlukan (Juta Pound) Zat khusus 60 Sulfur oksida 150 Hidrokarbon 125 Reduksi laju emisi (Jutaan pound) dari metode pengurangan yg dilakukan Nori&Leets Co. Polutan Tambah tinggi cerobong asap (1) Menggunakan alat penyaring di cerobong (2) Total biaya yang dapat digunakan utk pengurangan (Juta dolar) Metode Pengurangan Menggunakan bahan bakar yg lebih baik (3) BF OF BF OF BF OF Zat khusus Sulfur oksida Hidrokarbon BF OF (1) 8 10 (2) 7 6 (3) 11 9

21 LP: Exercises Selesaikan dengan menggunakan metode grafik persoalan berikut: Maximize Z = 2x 1 + x 2 Subject to: x 1 + 2x 2 15 x 1 + 2x x 1 + 3x 2 45 x 1 0 x 2 0

22 Lecture 3 - Preparation Read and Practice: Hamdy A. Taha. Operations Research: An Introduction. 8th Edition. Prentice-Hall, Inc, Page: (Examples to 2.3.3).

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